4 Trigonometric Functions Copyright Cengage Learning All rights

4 Trigonometric Functions Copyright © Cengage Learning. All rights reserved.

4. 8 Applications and Models Copyright © Cengage Learning. All rights reserved.

What You Should Learn • Solve real-life problems involving right triangles. • Solve real-life problems involving directional bearings. • Solve real-life problems involving harmonic motion. 3

Applications Involving Right Triangles 4

Applications Involving Right Triangles Instructions for setting up your triangle: The three angles of a right triangle are denoted by the letters A, B and C (where C is the right angle), and the lengths of the sides opposite these angles by the letters a, b and c (where c is the hypotenuse). 5

Example 1 – Solving a Right Triangle Solve the right triangle shown in Figure 4. 77 for all unknown sides and angles. Solution: Figure 4. 77 Because C = 90 , it follows that A + B = 90 and B = 90 – 34. 2 = 55. 8. 6

Example 1 – Solution cont’d To solve for a, use the fact that a = b tan A. So, a = 19. 4 tan 34. 2 13. 18. Similarly, to solve for c, use the fact that 23. 46. 7

Trigonometry and Bearings 8

Trigonometry and Bearings In surveying and navigation, directions are generally given in terms of bearings. A bearing measures the acute angle a path or line of sight makes with a fixed north-south line, as shown in Figure 4. 81. For instance, the bearing of S 35 E in Figure 4. 81(a) means 35 degrees east of south. (a) (b) Figure 4. 81 (c) 9

Example 5 – Finding Directions in Terms of Bearings A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P. M. the ship changes course to N 54 W, as shown in Figure 4. 82. Find the ship’s bearing and distance from the port of departure at 3 P. M. Figure 4. 82 10

Example 5 – Solution For triangle BCD you have B = 90 – 54 = 36 . The two sides of this triangle can be determined to be b = 20 sin 36 and d = 20 cos 36 . In triangle ACD, you can find angle A as follows. 0. 2092494 11

Example 5 – Solution cont’d A arctan 0. 2092494 0. 2062732 radian 11. 82 The angle with the north-south line is 90 – 11. 82 = 78. 18 . 12

Example 5 – Solution cont’d So, the bearing of the ship is N 78. 18 Finally, from triangle ACD you have which yields 57. 39 nautical miles Distance from port 13

Harmonic Motion 14

Harmonic Motion The periodic nature of the trigonometric functions is useful for describing the motion of a point on an object that vibrates, oscillates, rotates, or is moved by wave motion. For example, consider a ball that is bobbing up and down on the end of a spring, as shown in Figure 4. 83 15

Harmonic Motion Suppose that 10 centimeters is the maximum distance the ball moves vertically upward or downward from its equilibrium (at-rest) position. Suppose further that the time it takes for the ball to move from its maximum displacement above zero to its maximum displacement below zero and back again is t = 4 seconds. Assuming the ideal conditions of perfect elasticity and no friction or air resistance, the ball would continue to move up and down in a uniform and regular manner. 16

Harmonic Motion From this spring you can conclude that the period (time for one complete cycle) of the motion is Period = 4 seconds its amplitude (maximum displacement from equilibrium) is Amplitude = 10 centimeters and its frequency (number of cycles per second) is Frequency = cycle per second. Motion of this nature can be described by a sine or cosine function, and is called simple harmonic motion. 17

Harmonic Motion 18

Example 6 – Simple Harmonic Motion Write the equation for the simple harmonic motion of the ball illustrated in Figure 4. 83, where the period is 4 seconds. What is the frequency of this motion? Figure 4. 83 19

Example 6 – Solution Because the spring is at equilibrium (d = 0) when t = 0, you use the equation d = a sin t. Moreover, because the maximum displacement from zero is 10 and the period is 4, you have the following. Amplitude = | a | = 10 =4 Consequently, the equation of motion is d = 10 20

Example 6 – Solution cont’d Note that the choice of a = 10 or a = – 10 depends on whether the ball initially moves up or down. The frequency is Frequency cycle per second. 21